Asymptotic aspects of the Boussinesq approximation for gases and liquids

Abstract

The asymptotic formulation of the Boussinesq approximation relates the pressure of the fluid to a thermodynamical quantity involving the heat capacity c_Po . In this paper we examine the implications of such a scaling, in particular: (i) the singular degeneracy of the equation of state ρ = ρ* (1 ?α* (T?;T*)) of a liquid: this equation of state is valid only for small values of the coefficient α T*; (ii) in which manner the scaling introduces the Mach number of the flow as a small parameter e for a compressible fluid. The equations at order zero with respect to ? are the same equations for gases and for liquids only if the thermodynamics of the medium is described by using the Brunt-Väisälä frequency instead of the temperature.